The Study Of Estimator Comparison And Prediction Theory In Linear Model

Linear model is a kind of important model in modern statistics, and is widely used in many areas such as economics, finance and so on. During the modeling analysis of linear model, the theory of parametric estimation is very important and is attached great importance by statisticians. On the one hand, statisticians study the theories and methods of parametric estimation, and analyze the dominance properties of various es-timators. On the other hand, they make use of the methods of parametric estimation to predict the future observations. In this context, we mainly study the dominance prop-erties of various parametric estimators in linear model, and the prediction methods of finite population regression coefficient based on the statistical decision theories.In the linear model with multivariate t errors, we investigate the dominance proper-ties of Stein-rule (SR) estimator, Positive-part Stein-rule (PSR) estimator, feasible min-imum mean squared error (FMMSE) estimator, and adjusted minimum mean squared (AFMMSE) estimator for regression coefficient under balanced loss function. Firstly, we give the unified expression for four estimators by the idea of pre-test estimator and derive the explicit formulae of the risks for these estimators. Secondly, based on the exact risks, we analyze the dominance properties of PSR estimator and SR estimator. Finally, in view of the complication of risks, we further compare these estimators by numerical analysis.In the linear model with elliptically contoured errors, we study the performance of least squared estimator (LSE), restricted least squared estimator (RE), preliminary test estimator (PTE) and Stein-type estimator (SE) for the error variance under squared error loss. Firstly, we obtain the exact risks of four estimators based on the properties of elliptically contoured distribution. On the basis of this, the influence factors of the risk size of PTE are analyzed. Meanwhile, we give the relation between the risk of PTE and that of LSE, and RE. Moreover, we discuss the optimal critical value of PTE. Secondly, we further compare four estimators by numerical analysis and bootstrap method in a special case that the error term of the model obeys a multivariate t distribution, since the risks are complicated and depend on the unknown parameters.For the prediction problem of finite population regression coefficient, under a bal-anced loss function, we investigate the admissible predictor in superpopulation models with and without the assumption that the underlying distribution is normal, respectively. In the non-normal case, we obtain the necessary and sufficient conditions for linear pre-dictors to be admissible in the class of homogeneous linear predictors. Meanwhile, we give the best linear unbiased predictor (BLUP) for finite population regression coeffi-cient and analyze the admissibility of BLUP in the class of homogeneous linear pre-dictors. In the normal case, we discuss the problem whether the homogeneous linear admissible predictors is admissible in the class of all predictors. Firstly, we obtain the sufficient conditions for homogeneous linear admissible predictor to be admissible in the class of all predictors. Secondly, we prove the sufficient conditions are also nec-essary under additional conditions. Finally, we give the best unbiased predictor (BUP) for finite population coefficient, and analyze the admissibility of BUP in the class of all predictors.Finally, under an adjusted balanced loss function, we investigate the Minimax pre-dictor for finite population regression coefficient in the superpopulation models with and without the assumption that the underlying distribution is normal, respectively. In the non-normal case, we obtain linear Minimax predictor of finite population regression coefficient in the class of homogeneous linear predictors. Meanwhile, we discuss the admissibility of the linear Minimax predictor in the class of homogeneous linear pre-dictors, and compare it with the best linear unbiased predictor proposed by Bolfarine. In the normal case, we study Minimax predictor of finite population regression coeffi-cient in the class of all predictors. Meanwhile, we analyze the admissibility of Minimax predictor in the class of all predictors and compare it with simple projection predictor.